10$~GeV on the scattered electron, and a cut on the pseudo-rapidity $\eta=-\ln\tan(\theta/2)$ of the scattered lepton and jets of $|\eta|<3.5$. First studies of the scale dependence of the dijet cross section in the cone scheme are presented in \cite{mz1}. We have considered scales related to the scalar sum of the parton transverse momenta in the Breit frame, $\sum_i \,p_T^B(i)$, and the virtuality $Q^2$ of the incident photon. In the following we will also consider scales related to $\sum_i \,k_T^B(i)$. Here, $(k_T^{B}(i))^2$ is defined by $2\,E_i^2(1-\cos\theta_{iP})$, where the subscripts $i$ and $P$ denote the final parton (or jet) and proton, respectively (all quantities are determined in the Breit frame). %The Breit frame is characterized by the property that %the momentum of the exchanged photon $q_\mu$ is completely spacelike, %having nonzero energy component The Breit frame is characterized by the vanishing energy component of the momentum of the exchanged photon. Both the photon momentum \begin{equation} q=(0,0,0,-2xE),\,\,\,-q^2=Q^2=4x^2E^2\\ \end{equation} and the proton momentum \begin{equation} P=E(1,0,0,1) \end{equation} are chosen along the $z$-direction. $x$ is the standard Bjorken scaling variable. In the parton model, the incoming quark with momentum $p=xE(1,0,0,1)$ collides elastically with the virtual boson and is scattered in the opposite direction (the 'current hemisphere') with momentum $p'=xE(1,0,0,-1)$; therefore, $(k_T^{B}(p'))^2= Q^2$ in the limit of the quark parton model, whereas $p_T^B(p')=0$. The kinematics for dijet production is more complex: the momentum fraction $\eta$ of the incoming parton must be larger than $x$ since $m_{jj}^2= Q^2(\eta/x-1)$ and, in general, the jets have a nonvanishing transverse momentum with respect to the $\gamma^*$-proton direction. At LO, i.e. for massless jets, the relation between $k_T^B(j)$ and $p_T^B(j)=E_j\sin\theta_{jP}$ reads: \begin{equation} k_T^B(j) = p_T^B(j)\sqrt{\frac{2}{1+\cos\theta_{{\,\!jP}}}} \end{equation} Obviously, $k_T^B(j)>p_T^B$, and one can also show that $\sum_j \,k_T^B(j) > Q$. Thus, $\sum_j \,k_T^B(j)$ is approximately given by the harder of the two scales $Q$ and $\sum_j \,p_T^B(j)$ \cite{mz3}. For large dijet invariant masses (i.e. for ``true'' two jet kinematics) one has $\eta >> x$, the dijet system will be strongly boosted in the proton direction and $k_T^B(j) \approx p_T^B(j)>>Q/2$, as long as very small scattering angles in the center of mass frame are avoided. For $m_{jj}\approx Q$ or smaller, on the other hand, $\sum_j \,k_T^B(j)\approx Q$ and, typically, both are considerably larger than $\sum_j \,p_T^B(j)>m_{jj}$, which corresponds to the parton model limit). Thus $\sum_j \,k_T^B(j)$ smoothly interpolates between the correct limiting scale choices, it approaches $Q$ in the parton limit and it corresponds to the jet transverse momentum when the photon virtuality becomes negligible. It appears to be the ``natural'' scale for multi jet production in DIS. In \cite{mz1} we found that the scale dependence of the dijet cross section does not markedly improve in NLO for $\mu^2=\xi Q^2$. This is also shown in Fig.~1 (dotted curves) where the dependence of the two-jet cross section on the scale factor $\xi$ is shown. We used the parton distribution functions set MRS D-$^\prime$ \protect\cite{mrs} and employed the two loop formula for the strong coupling constant both in the LO and NLO curves. For scales related to $\sum_i \,p_T^B(i)$ the uncertainty from the variation of the renormalization and factorization scale is markedly reduced compared to the LO predictions (dashed curves in Fig.~1). %For scales related to $\sum_i \,p_T^B(i)$, Here $\xi$ is defined via % % \begin{equation} \mu_R^2 = \mu_F^2 = \xi\;(\sum_i \,p_T^B(i))^2\,. \label{xidef} \end{equation} % % %and similar for $(\sum_i \,k_T^B(i))^2$. The resulting $\xi$ dependence for $ \mu_R^2 = \mu_F^2 = \xi\;(\sum_i \,k_T^B(i))^2$ is shown as the solid lines in Fig.~1. In this case, the NLO two-jet cross section is essentially independent on $\xi$ for $\xi<2$. Hence, the theoretical uncertainties due to the scale variation are very small suggesting a precise determination of $\alpha_s( )$ for different $ $ bins, where \begin{equation} =\frac{1}{2}\,\, (\sum_{j=1,2} \,k_T^B(j)) \label{akt} \end{equation} % % \begin{figure}[tp] \vspace{5.5cm} \begin{picture}(7,7) %\special{psfile=scale.ps voffset=-150 hoffset=-10 hscale=40 \special{psfile=scale1.ps voffset=-170 hoffset=-80 hscale=60 vscale=60 angle=0} \end{picture} \caption{ Dependence of the two-jet exclusive cross section in the cone scheme on the scale factor $\xi$. The dashed curves are for $\mu_R^2=\mu_F^2=\xi\;(\sum_i\;p_T^B(i))^2$. Choosing $(\sum_i\;k_T^B(i))^2$ as the basic scale yields the solid curves. Choosing $Q^2$ as the basic scale yields the dotted curves. Results are shown for the LO (lower curves) and NLO calculations. } \end{figure} \begin{figure}[tp] \vspace{5.5cm} \begin{picture}(7,7) %\special{psfile=scale.ps voffset=-150 hoffset=-10 hscale=40 \special{psfile=fig_kt.ps voffset=-170 hoffset=-80 hscale=60 vscale=60 angle=0} \end{picture} \caption{ $ $ distribution for the two-jet exclusive cross section. The parameters are explained in the text. } \end{figure} \begin{figure}[tp] \vspace{5.5cm} \begin{picture}(7,7) %\special{psfile=scale.ps voffset=-150 hoffset=-10 hscale=40 \special{psfile=fig_pt.ps voffset=-170 hoffset=-80 hscale=60 vscale=60 angle=0} \end{picture} \caption{ $ $ distribution for the three $ $ bins shown in Fig.~1. } \end{figure} \begin{figure}[tp] \vspace{5.5cm} \begin{picture}(7,7) %\special{psfile=scale.ps voffset=-150 hoffset=-10 hscale=40 \special{psfile=fig_qq.ps voffset=-170 hoffset=-80 hscale=60 vscale=60 angle=0} \end{picture} \caption{ $Q^2$ distribution for the three $ $ bins shown in Fig.~1. } \end{figure} Fig.~2 shows the $ $ distribution for the NLO exclusive dijet cross section. We used the parton distribution functions set GRV \cite{grv} and $\mu_R^2=\mu_F^2=1/4\;(\sum_i\;k_T^B(i))^2$. In addition to the cuts imposed in Fig.~1, we require $p_T^B>4$~GeV for each jet %and $k_T^B>4$ GeV in the Breit frame, and $\sum_j k_T^B(j)> 10$ GeV. We have divided the NLO cross section in the following three $ $ bins.\\ bin 1: 5~GeV $<\,\,\,\, \,\, \,\,<$ 10~GeV \\ bin 2: 10~GeV $<\,\,\,\, \,\,\,\,<$ 15~GeV \\ bin 3: 15~GeV $<\,\,\,\, $.\\ Table~1 shows the corresponding NLO cross sections for these three bins for two different sets of parton distributions and for different values for the scale factor $\xi$. The theoretical uncertainties for the NLO dijet cross section from these numbers are very small, in particular for the first two bins. \begin{table}[t] \caption{Jet cross sections in pb for the three $ $ bins. }\label{table1} \vspace{2mm} \begin{tabular}{lccc} \hspace{0.8cm} & \mbox{bin 1 } & \mbox{bin 2 } & \mbox{bin 3 } \\ \hline\\[-3mm] \mbox{GRV} $\xi=1/4 $ & 497 & 320 & 146 \\ \mbox{GRV} $\xi=1/16$ & 504 & 306 & 134 \\ \mbox{GRV} $\xi=1 $ & 488 & 322 & 151 \\ \mbox{MRSD-'} $\xi=1/4 $ & 487 & 322 & 148 \\ \end{tabular} \end{table} Fig.~3 and 4 show the $ $ and $Q^2$ distribution for the NLO exclusive dijet cross section for these three bins. Whereas the $ $ and $Q^2$ distributions are fairly similar to the $ $ distribution for the lowest bin, large differences are found for the other bins. This partly explains the rather different scale dependence observed in Fig.1. \section{Forward Jet Production in the Low $x$ Regime} Deep inelastic scattering with a measured forward jet with relatively large momentum fraction $x_{jet}$ (in the proton direction) and $p_T^{2\,lab}(j)\approx Q^2$ is expected to provide sensitive information about the BFKL dynamics at low $x$ \cite{mueller,allen1}. In this region there is not much phase space for DGLAP evolution with transverse momentum ordering, whereas large effects are expected for BFKL evolution in $x$. In particular, BFKL evolution is expected to substantially enhance cross sections in the region $x< 8~$GeV$^2$ , $x<0.004$, $0.1 < y < 1$, an energy cut of $E(e^\prime)>11$~GeV on the scattered electron, and a cut on the pseudo-rapidity $\eta=-\ln\tan(\theta/2)$ of the scattered lepton of $ -2.868 < \eta(e^\prime)< -1.735$ (corresponding to $160^o < \theta(l^\prime) < 173.5^o$). Jets are defined in the cone scheme (in the laboratory frame) with $\Delta R = 1$ and $|\eta(j)|<3.5$. We require a forward jet with $x_{jet}=p_z(j)/E_{P} > 0.05$, $E(j)>25$ GeV, $0.5 0.05$) and restricted transverse momentum ($0.5 0.05$ requirements. For $x< > Q^2\; , \end{eqnarray} where $m_T$ and $y$ are the transverse mass and rapidity of the partonic recoil system, respectively. Thus a recoil system with substantial transverse momentum and/or invariant mass must be produced and this condition favors recoil systems composed out of at least two additional energetic partons. As a result one finds very large fixed order perturbative QCD corrections (compare 2 jet LO and NLO results with a forward jet in Table~\ref{table2}). In addition, the LO $({\cal{O}}(\alpha_s^2))$ 3-jet cross section is larger than the LO $({\cal{O}}(\alpha_s))$ 2-jet cross section. Thus, the forward jet cross sections in Table~\ref{table2} are dominated by the $({\cal{O}}(\alpha_s^2))$ matrix elements. The effects of BFKL evolution must be seen and isolated on top of these fixed order QCD effects. We will analyze these effects in a subsequent publication. \section{Conclusions} The calculation of NLO perturbative QCD corrections has received an enormous boost with the advent of full NLO Monte Carlo programs \cite{giele1,jim}. For dijet production at HERA the NLO Monte Carlo program MEPJET \cite{mz1} allows to study jet cross sections for arbitrary jet algorithms. Internal jet structure, parton/hadron recombination effects, and the effects of arbitrary acceptance cuts can now be simulated at the full ${\cal O}(\alpha_s^2)$ level. We found large NLO effects for some jet definition schemes (in particular the $W$-scheme) and cone and $k_T$ schemes appear better suited for precision QCD tests. The extraction of gluon distribution functions is now supported by a fully versatile NLO program. 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